Abstract

Hyperthermia, the heating of cancerous tumors can improve the efficiency of cancer treatment when added as an adjuvant to Radio therapy and Chemotherapy. Main goal of this work is to design a periodic output feedback controller to control the power of ultrasound transducer so that uniform temperature distribution is maintained in the targeted tissues all through the hyperthermia treatment process. For this purpose first a tumor layer surrounded by muscle layer is modeled using Penne’s bio heat transfer equation and a higher order state space model to the bio heat transfer problem is obtained by finite difference method. In second step an experimental proto type for the above model is created using agar phantom which mimics the human tissue. Temperature response of the simulated model is compared with the experimental outcome to demonstrate the validity of simulated model. Now a periodic output feedback controller is designed for the simulated hyperthermia model. The performance of the controller is evaluated by framing a desired trajectory which meets the treatment goals of hyperthermia. Closed loop error norm and the open loop error norm are evaluated to prove the efficiency of the designed controller. Simulations are also done to show the robustness of the controller in spite of variation in blood perfusion. Simulations proved that hyperthermia system is robust to blood perfusion variation and the closed loop norm has improved more than 40% compared to open loop norm in certain perfusion cases.

Keywords

Introduction

Cancer is a major hazard to human life. Hyperthermia the
heating of cancerous tumors, can improve response rates
when added as an adjuvant treatment to radiation therapy.
Recent experiments on human subjects confirm that in
cervical cancer and recurrent lesions of malignant melanoma
the response rate is 53% and 28% for patients who
received radiation only and the response rate has improved
to 83% and 46% respectively for patients who
received radiation in adjuvant with hyperthermia. Reoxygenation
of the hypoxic tumor regions occur and become
radiosensitive, when tumor temperature is raised to
40°C - 44°C for 30-60 min .The prime goal of an online
hyperthermia controller is to maintain desired temperature
of 43°C within the tumor while limiting temperatures outside
the tumor to safe levels of < 40 °C from a baseline
body temperature of 37°C [1].This goal must be met under
the influence of variable blood perfusions, that lead to
plant model mismatch [2]. Several control algorithms
have been suggested for this purpose.

Formerly many researchers have developed automatic
temperature controllers for various hyperthermia systems
such as scanned focused & phased array ultrasound.
Many control ideas in the range of basic PID [3], LQR
controller [4], multipoint adaptive, more complex model
predictive [2], Constrained Predictive Control [5], Minimum-
Time Thermal Dose Control [6] and even recursive
control procedure [7] were used to design the system. Due
to complexity of human body and finite difference modeling
the full order models developed were too large. Potocki
and Tharp [8], Mattingly and Romer [1] and Marshal
et. al [9] reduced the order of the hyperthermia system
and designed controllers. But none of the designed
controllers relay on reduced models for designing controller
for higher order system, Lamba and Rao [10] showed
that if a state feedback control is designed from the Davison
[11] reduced model and applied to the higher order
system it results in stable closed loop system but this necessitates
the system state to be available for feedback.

In hyperthermia, it is essential to retain a homogeneous
temperature in the tissue and tumor. If the distribution of temperatures within the tumor shows considerable dispersion
about it mean value, treatment failures and complications
are liable to occur i.e cold spots never reach therapeutic
temperatures and so are not treated. Similarly hot
spots lead to complications such as unbearable pain and
ulceration of the overlying skin. So to have uniform temperature
distribution it is sensible to measure the temperature
at each node. As in hyperthermia all system states are
not accessible for feedback it insists the need for estimator
design, this makes the system complex and reduces
system reliability. Hence, it is rather attractive to go for an
output feedback design. The output feedback needs only
the measurement of system output in contrast, to the state
feedback which requires the knowledge of the states or a
state estimator. In such a situation it is better to use periodic
output feedback technique which requires the feedback
of the output temperature signal only. Comparison to
other techniques periodic output feedback is much simpler.
The control signal is kept constant over few measurements
and altered periodically.

Although the existing controllers are capable of satisfying
the basic needs of on line hyperthermia system they claim
for the need of estimator and the reduced order controller
does not guarantee the closed loop stability for the higher
order system. So in the proposed method an output feedback
controller is designed for higher order discrete system
via reduced order model. The benefits of the method
are two folds, first the states of the system need not be
estimated and secondly when the controller is placed in
the closed loop with the higher order it guarantees the
closed loop stability. This study is a first effort towards
including an output feedback controller for hyperthermia
system.

Materials

Tumor & Ultrasonic Field Modeling

A simple 1-D inhomogeneous tissue is modeled as a tumor
layer surrounded by muscle layer on either side as in Figure
1.

Arterial temperature Ta is assumed to be 37°C and Qa is
the power deposited in the ultrasonically heated tissue.
The heterogeneous tissue with tumor modeled in 1-D using
Penne’s bio heat equation contains conduction effects,
geometrical information on normal and diseased tissue.
The parameter Wb characterizes the energy eliminated by
conduction in the plane at right angles to the ultrasound
axis.Table-1 summarize the thermal properties of human
tissue.

Table 1: Thermal properties of human tissue.

Thermal conductivity is assumed to be constant and ‘x’ is
the depth of penetration inside the tissue. In our model x
varies from 0 to 13 cm and the boundary condition are
assumed to be T (0, t) = Ta and T (L, t) = Ta. This is due
to thermoregulation effect. In this paper we assume a single
scanned focused ultrasound transducer with fixed
characteristics .The power deposition term Qa is modelled
as the energy deposited by scanned focused ultrasound
transducer and given

(2)

Where Qi (x), αi and Si are energy deposition, attenuation
co-efficient and penetration length for layer ’r’ The transducer
is positioned 17cm from the front edge of the tissue
with r=25cm, d=70mm, l =1mm. Where I(0) is the average
intensity over the radiating surface ‘d’ is the diameter
of the transducer ’r’ is the radius of curvature and ‘x’ is
the distance from the centre of the transducer

State Space Formulation

The numerical method finite difference is used for solving
the partial difference equation of bio heat transfer.

(3)

Assume is the elevated temperature, where represents the temperature in the tissue and Ta arterial
temperature.

(4)

The tissue model is split into 131 finite difference nodes
.The temperature distribution in each node is known
based on equation (4) and power deposition in each node
is given according to equation (2).

The system matrix A is a higher order tri-diagonal matrix
with non zero elements only on its main diagonal and upper
and lower diagonals. and symbolizes the tri-diagonal elements of normal tissue and signify the tri-diagonal elements of tumor tissue respectively.
Here c1and c2 are tumor tissue specific heat capacity
and normal tissue specific heat capacity respectively

(5)

The value Q in the input matrix B represents the spatial
power deposition at each finite difference node. The output
matrix C of the model is represented by the location
of the sensor.

The value Q in the input matrix B represents the spatial
power deposition at each finite difference node. The output
matrix C of the model is represented by the location
of the sensor.

Where A is a tri diagonal system matrix incorporating
both conduction and perfusion terms and B is the input
matrix. For 1-D case the state T is a vector of temperatures
elevation in the nodes of the tissue model and U is
I(0) a single manipulated variable. The position of the
ultrasound transducers is fixed and the magnitude of the
ultrasound power is the only manipulated variable.

Experimental Prototype

An experimental set up similar to that used by Arora et.al
[13] was used for model validation and control experiments
and is shown in Figure-2. An 1-D cylindrical agar
phantom of length 13 cm and diameter of 1.27 cm was
considered. Condensed milk was added to mimic the ultrasound
absorption properties of human tissue [14]. Ultrasound
field was generated by single focused ultrasound
transducer resonating at a frequency of 2 MHz. The radius
of curvature and diameter of the transducer are 25cm and
7cm respectively. The agar phantom and the ultrasound
transducer are immersed in a bath of degassed water to
simulate the cooling effect of blood and to avoid the disturbance
caused by air bubbles. The RF signal generated
by a RF signal source and RF amplifier were used to drive
the transducer.

Figure 2.Temperature measurement in I-D Agar Phantom

The temperature in the agar phantom was measured by a
manganin-constantan thermocouple fixed on it. These
thermocouples give an accuracy of ± 0.2 °C in temperature
measurement over the range of 35°C -55°C, thereby
eliminating the errors caused by conduction. Signal conditioning
circuits and A/D cards are used to acquire and
monitor the temperature.

Methods

Control Objective and Constraints

For modeling and simulation purpose the desired temperature
was fixed to 43°C within the tumor while limiting
the temperature outside the tumor to safe levels of
less than 40°C .The objective of the control algorithm is
to track the step variations of the power input without
overshoot, with a rise time varying between 6 min and 12
min (6 min < tr < 12 min) and to have a maximum settling
time of 12 min. In addition to the above the constraint on
the control signal is limited to Umax. Usually the maximum
allowed power deposition is determined by hardware
constraints or to avoid cavitations in tissues [2].

Periodic Output Feedback Control Technique

In periodic output feedback technique the input is
changed several times in one output sampling instant .The
poles of a controllable and observable system discretized
at output sampling rate can be arbitrarily assigned by a
piecewise constant periodic output feedback provided the
number of gain changes during one output sampling interval
[15,16,17].

The system after sampling by Γ secs

(6)

Assume (ΦΓ
,C) F is observable and (F ,G) G controllable
with controllability indexg . Now the output sampling
interval is divided into N subintervals of length Δ=Γ/N
such that N ³ g . Thus the control law for POF controller is
given as

(7)

The sequence of N gains generate a
time varying piecewise constant output feedback gains K(t) for 0 ≤ t ≤t .Sampling the simulated system by the
sampling interval Δ, the delta sampled system is represented
a

(8)

Define

(9)

A stabilizing output injection gain G0 is designed such that
the Eigen values of lie inside the unit circle.
The periodic output gain K is calculated solving

ΓK=G0 (10)

Where

The closed loop dynamic of the system with POF gain
can be given as

(11)

The POF controller obtained by this method requires only
constant gains and hence easy to implement. A controller
designed as above will give desired closed loop behavior,
but strong oscillations exists between sampling instants
.Werner proposed a performance index such that ΓK » Γ0
and ΓK=G0 need not be imposed [18]. This constraint is
replaced by a penalty function, which makes it possible to
enhance the closed loop performance, by permitting slight
variation from the original design and at the same time
improving the inter sample behavior. A tradeoff between
the closed loop performance and the similarity to the chosen
design is expressed by the quadratic performance index
as stated below is minimized

(12)

The state and input of the system are characterized as xl,
ul, respectively. represents the state that would be
reached at the instant k=lN, given if k is solved to
satisfy GK = G exactly. Q, R and P are positive definite
symmetric matrices of appropriate dimensions. The first
term represents the state and the control energy of the
system and second term represents the penalty for the
deviation of the output injection gain from exact value
‘Go’. The POF gain calculated thus have very low magnitude
and is represented by K1 [19].

Desired Trajectory

A function which gives the desired trajectory for temperature
evaluation at each point in tumor and in normal
healthy tissues is framed from the control objective of the
treatment protocol. For each perfusion case the performance
analysis is done by calculating the error norm. The
error norm is calculated as the 2-norm of the difference
between the desired trajectory and the achieved output
trajectories.

Result and Discussions

System Simulation & Model Validation

Temperature predictions from the simulated model are
validated with the measured temperature from the temperature
measured by the manganin-constantan thermocouple
and is shown in Figure -3. A step increase in ultrasound
power of 7 watts was applied prior to the treatment
session. To validate the simulated model with experimental
reading the temperature is measured at a single spatial
location at a distance of 2cm from the edge of the phantom
by fixing a thermocouple. Total of 200 temperature
data samples are recorded with the sampling time of 5
secs. The time temperature response at 20th node from the
simulated mathematical model is also drawn and both
were compared. Figure 3 shows the comparison between
the experimental output and simulated output. This shows
an accurate steady state prediction and there is a small
error in the time constant of the response. The reason for
this error is the Bio-heat transfer equation used for simulating
the model is 1-D where as the agar phantom used is
of diameter 1.25, the heat conduction in lateral direction
would have contributed for the plant model mismatch
created the error.

Space discretization of the model gives rise to 131 nodes
including the boundary nodes. During hyperthermia blood
perfusion is the major variable that leads to parameter
variation. Perfusion conditions applied in simulation are
given as tumor perfusion WT and normal tissue perfusion
WN. The typical values WT and WN used in hyperthermia
system modeling ranges from a lower extreme of 0.5
kg/m3sec to a higher extreme of 10 kg/m3sec[20] . Four
cases are considered with different combinations of WT
and WN in this range, also simulations are also carried out
by varying thermocouple locations. The open loop time
temperature response is obtained by heating the tissue
tumor model with constant applied power for 60min.
Since it is a temperature process ‘τ 'is chosen to be
12secs. The original system is of 131 x131 orders. Since the system is too large for controller design the output
feedback controller design for the original system is done
via a reduced model of order 4. The number of gain
changes in one sampling interval is fixed as N = 4. So
Δ = Γ / N = 3sec s .

The response of the system in case-1 is shown in Figure 4,
The open loop step response, the closed loop response with
output injection gain G0, the desired trajectory that meets the
primary goal of hyperthermia are shown in Figure-4(a).The
closed loop performance of POF controller after optimization
procedure is in Figure-4(b) and the magnitude of the
control effort needed to achieve this response is in Figure
4(c).The system dynamics without controller is not satisfactory
and it does not meet the requirements of hyperthermia
cancer treatments. The rate of temperature rise is very fast
and this may lead to patient discomfort and pain stimulation.
The prolonged settling time may cause hot spots and
cold spots inside the tumor leading to decrease in treatment
efficiency .This necessitates the need for a controller
.The closed loop response with POF controller closely
fallow the desired trajectory This will reduce the error
norms of the closed loop system. The closed loop response
with POF controller satisfies the requirements of
standard hyperthermia. Although the simulations are done for 60mins the time axis is limited to 1000secs to show
the effectiveness of the controller. Simulations are done
for cases 2, 3 and 4 also but response is shown only for
case-1.

Figure 4:Temperature response for the perfusion case-i
(WT=0.5 kg/(m3sec) and WN =0.5 kg/(m3sec). with Thermocouple
at normal tissue location (a) open loop and closed
loop response with state feedback and desired trajectory (b)
Closed loop response with Periodic Output Feedback Controller
and desired trajectory (c) Control input as a function
of time

The control structure can be varied by changing the thermocouple
location. Now the thermocouple location is
changed to X=50 i.e. in tumor tissue. According to the
goals of hyperthermia temperature with in tumor tissue
should be ≥43°C. Figure-5 shows the response for case-I
with thermocouple location at X=50.

Figure 5:Temperature response for the perfusion case-i
(WT=0.5 kg/(m3sec) and WN =0.5 kg/(m3sec). with Thermocouple
at tumor tissue location (a) openloop and closedloop
response with state feedback and desired trajectory
(b) Closed loop response with Periodic Output Feedback
Controller and desired trajectory (c) Control input
as a function of time

Table 2 shows that the inclusion of piecewise constant
periodic output feedback controller makes the system
closed loop stable and also the norm of the error between
the desired and the achieved trajectories for the open loop
and the closed loop system. The shapes of the achieved
trajectories show maximum consistency and the calculated
error norms do not show significant deviation. The
closed loop error norm is much less than the open loop
error norm indicating the performance of the designed
controller. Since the power deposition is more in the tumor
region variation in tumor perfusion WT leads to a
larger effect on the temperature field. To overcome the
slow dynamics of the system higher control efforts are
needed at the beginning. For a given power input decrease
in blood perfusion causes an increase in system temperature.
Table 3 shows the norm of error between the desired
and achieved trajectories for varying blood perfusion in
tumor tissue.

Table 2: Openloop and closed loop error norms with
POF Controller for varying blood perfusion and the
thermocouple located at normal tissue

Table 3: Open loop and closed loop error norms with POF Controller for varying blood perfusion and the
thermocouple located at tumor tissue

Conclusion

Model validation results prove that the simulated model is
a good approximation to the experimental prototype.
Closed loop norms demonstrate the effectiveness of controller
in achieving the goals of hyperthermia. It is found
that the stabilizing controller designed from the reduced
order model and applied to the higher order system performance
and stability is guaranteed. Periodic output
feedback controller developed is flexible and can be applied
to hyperthermia models with changing blood perfusion
leading to plant model mismatch. The suppression of
fluctuations avoids hot spots and cold spots in tumor and
surrounding tissues. The control effort is reduced by
quadratic performance index optimization this avoids the controller saturation. For smaller perfusion rates the system
settling time is longer. As the blood perfusion increases,
it takes more heat away from the tissues and
more control effort is needed to maintain a steady temperature.
The output feedback controller effectively adjusts
the power level of the ultrasound transducer according
to the blood perfusion and other dynamic tissue properties
to achieve controlled effective ultrasound hyperthermia.
The results demonstrate that the periodic output
feedback controllers are able to effectively track the target
temperature with dynamic tissue properties. Achieving
the desired trajectory leads to treatment precribability and
this gives clinical acceptance to hyperthermia.

References

Marshall Mattingly, Robert BR, Santhosh Devasia Exact temperature Tracking for hyperthermia: A Model Based Approach IEEE Transactions on control system Technology 2000; 8(6):979-992.